"Sit down, Pyne," said Vance, with peremptory kindness. "We have permission from Professor Dillard to question you; and we shall expect answers to all our questions."

    "Certainly, sir," the man answered. "I'm sure there's nothing that Professor Dillard has any reason to hide."

    "Excellent." Vance lay back lazily. "To begin with, then; what hour was breakfast served here this morning?"

    "At half past eight, sir—the same as always."

    "Were all the members of the family present?"

    "Oh, yes, sir."

    "Who calls the family in the morning? And at what time?"

    "I do myself—at half past seven. I knock on the doors—"

    "And wait for an answer?"

    "Yes, sir—always."

    "Now think, Pyne: did every one answer you this morning?"

    The man inclined his head emphatically. "Yes, sir."

    "And no one was late to breakfast?"

    "Every one was on time promptly—as usual, sir."

    Vance leaned over and deposited his cigarette ash in the grate.

    "Did you happen to see any one leaving the house or returning to it this morning before breakfast?"

    The question was put casually, but I noted a slight quiver of surprise in the butler's thin drooping eyelids.

    "No, sir."

    "Even though you saw no one," pursued Vance, "would it not have been possible for some member of the household to have gone out and returned without your knowing it?"

    Pyne for the first time during the interview appeared reluctant to answer.

    "Well, sir, the fact is," he said uneasily, "any one might have used the front door this morning without my knowing it, as I was in the dining-room setting table. And, for the matter of that, any one might have used the archery-room door, for my daughter generally keeps the kitchen door closed while preparing breakfast."

    Vance smoked thoughtfully a moment. Then in an even, matter-of-fact tone he asked: "Does any one in the house own a revolver?"

    The man's eyes opened wide.

    "Not that I—know of, sir," he answered haltingly.

    "Ever hear of the Bishop, Pyne?"

    "Oh, no, sir!" His face blanched. "You mean the man who wrote those letters to the papers?"

    "I merely meant the Bishop," said Vance carelessly. "But tell me: have you heard anything about a man being killed in Riverside Park this morning?"

    "Yes, sir. The janitor next door was telling me about it."

    "You knew young Mr. Sprigg, didn't you?"

    "I'd seen him at the house here once or twice, sir."

    "Was he here recently?"

    "Last week, sir. Thursday I think it was."

    "Who else was here at the time?"

    Pyne frowned as if trying to remember.

    "Mr. Drukker, sir," he said after a moment. "And, as I recall, Mr. Pardee came too. They were together in Mr. Arnesson's room talking until late."

    "In Mr. Arnesson's room, eh? Is it custom'ry for Mr. Arnesson to receive callers in his room?"

    "No, sir," Pyne explained; "but the professor was working in the library, and Miss Dillard was with Mrs. Drukker in the drawing-room here."

    Vance was silent a while.

    "That will be all, Pyne," he said at length. "But please send Beedle to us at once."

    Beedle came and stood before us with sullen aggressiveness. Vance questioned her along the same lines as he had taken with Pyne. Her answers, for the most part monosyllabic, added nothing to what had already been learned. But at the end of the brief interview Vance asked her if she had happened to look out of the kitchen window that morning before breakfast.

    "I looked out once or twice," she answered defiantly. "Why shouldn't I look out?"

    "Did you see any one on the archery range or in the rear yard?"

    "No one but the professor and Mrs. Drukker."

    "No strangers?" Vance strove to give the impression that the fact of Professor Dillard's and Mrs. Drukker's presence in the rear yard that morning was of no importance; but, by the slow, deliberate way in which he reached into his pocket for his cigarette-case, I knew the information had interested him keenly.

    "No," the woman replied curtly.

    "What time did you notice the professor and Mrs. Drukker?"

    "Eight o'clock maybe."

    "Were they talking together?"

    "Yes.—Anyway," she emended, "they were walking up and down near the arbor."

    "Is it custom'ry for them to stroll in the yard before breakfast?"

    "Mrs. Drukker often comes out early and walks about the flower beds. And I guess the professor has a right to walk in his own yard any time he wants to."

    "I'm not questioning his rights in the matter, Beedle," said Vance mildly. "I was merely wondering if he was in the habit of exercising those rights at such an early hour."

    "Well, he was exercising 'em this morning."

    Vance dismissed the woman and, rising, went to the front window. He was patently puzzled, and he stood several minutes looking down the street toward the river.

    "Well, well," he murmured. "It's a nice day for communin' with nature. At eight this morning the lark was on the wing no doubt, and—who knows?—maybe there was a snail on the thorn. But—my word!—all wasn't right with the world."

    Markham recognized the signs of Vance's perplexity.

    "What do you make of it?" he asked. "I'm inclined to ignore Beedle's information."

    "The trouble is, Markham, we can't afford to ignore anything in this case." Vance spoke softly, without turning. "I'll admit, though, that at present Beedle's revelation is meaningless. We've merely learned that two of the actors in our melodrama were up and about this morning shortly after Sprigg was snuffed out. The al-fresco rendezvous between the professor and Mrs. Drukker may, of course, be just one of your beloved coincidences. On the other hand, it may have some bearing on the old gentleman's sentimental attitude toward the lady. . . . I think we'll have to make a few discreet inquiries of him about his ante-prandial tryst, what? . . ."

    He leaned suddenly toward the window.

    "Ah! Here comes Arnesson. Looks a bit excited."

    A few moments later there was the sound of a key in the front door, and Arnesson strode down the hall. When he saw us he came quickly into the drawing-room and, without a word of greeting, burst forth:

    "What's this I hear about Sprigg being shot?" His eager eyes darted from one to the other of us. "I suppose you're here to ask me about him. Well, fire away." He threw a bulky brief-case on the centre-table and sat down abruptly on the edge of a straight chair. "There was a detective up at college this morning asking fool questions and acting like a burlesque sleuth in a comic opera. Very mysterious. . . . Murder—horrible murder! What did we know about a certain John E. Sprigg? And so on. . . . Scared a couple of juniors out of an entire semester's mental growth, and sent a harmless young English instructor into incipient nervous collapse. I didn't see the Dogberry myself—was in class at the time. But he had the cheek to ask what women Sprigg went around with. Sprigg and women! That boy didn't have a thought in his head but his work. Brightest man in senior math. Never missed a class. When he didn't answer roll-call this morning I knew something serious was the matter. At the lunch hour every one was buzzing about murder. . . . What's the answer?"

    "We haven't the answer, Mr. Arnesson." Vance had been watching him closely. "However, we have another determinant for your formula. Johnny Sprig was shot this morning with a little gun through the middle of his wig."

    Arnesson stared at Vance for some time without moving. Then he threw his head back and gave a sardonic laugh.

    "Some more mumbo-jumbo, eh?—like the death of Cock Robin. . . . Read me the rune."

    Vance gave him briefly the details of the crime.

    "That's all we know at present," he concluded. "Could you, Mr. Arnesson, add any suggestive details?"

    "Good Lord, no!" The man appeared genuinely amazed. "Not a thing. Sprigg . . . one of the keenest students I ever had. Something of a genius, by Gad! Too bad his parents named him John—plenty of other names. It sealed his doom apparently; got him shot through the head by a maniac. Obviously the same merry-andrew who did Robin in with an arrow." He rubbed his hands together,—the abstract philosopher in him had become uppermost. "A nice problem. You've told me everything? I'll need every known integer. Maybe I'll hit upon a new mathematical method in the process—like Kepler." He chuckled over the conceit. "Remember Kepler's 'Doliometrie'? It became the foundation of Infinitesimal Calculus. He arrived at it trying to construct a cask for his wine—a cask with a minimum amount of wood and a maximum cubical content. Maybe the formulas I work out to solve these crimes will open up new fields of scientific research. Ha! Robin and Sprigg will then become martyrs."

    The man's humor, even taking into consideration his life's passion for abstractions, struck me as particularly distasteful. But Vance seemed not to mind his cold-blooded cynicism.

    "There's one item," he said, "that I omitted to mention." Turning to Markham he asked for the piece of paper containing the formula, and handed it to Arnesson. "This was found beneath Sprigg's body."

    The other scrutinized it superciliously.

    "The Bishop, I see, is again involved. Same paper and typing as the notes. . . . But where did he get that Riemann-Christoffel tensor? Now, if it had been some other tensor—like the G-sigma-tau, for instance—any one interested in practical physics might have hit on it. But this one isn't common; and the statement of it here is arbitrary and unusual. Certain terms omitted. . . . By George! I was talking to Sprigg about this only the other night. He wrote it down, too."

    "Pyne mentioned the fact that Sprigg had called here Thursday night," put in Vance.

    "Oh, he did, did he? . . . Thursday—that's right. Pardee was here, too. And Drukker. We had a discussion on Gaussian co-ordinates. This tensor came up—Drukker mentioned it first, I think. And Pardee had some mad notion of applying the higher mathematics to chess. . . ."

    "Do you play chess, by the by?" asked Vance.

    "Used to. But no more. A beautiful game, though—if it wasn't for the players. Queer crabs, chess players."

    "Did you ever make any study of the Pardee gambit?" (At the time I could not understand the seeming irrelevance of Vance's questions; and I noticed that Markham too was beginning to show signs of impatience.)

    "Poor old Pardee!" Arnesson smiled unfeelingly. "Not a bad elementary mathematician. Should have been a high-school teacher. Too much money, though. Took to chess. I told him his gambit was unscientific. Even showed him how it could be beaten. But he couldn't see it. Then Capablanca, Vidman and Tartakower came along and knocked it into a cocked hat. Just as I told him they would. Wrecked his life. He's been fussing around with another gambit for years, but can't make it cohere. Reads Weyl, Silberstein, Eddington and Mach in the hope of getting inspiration."

    "That's most interestin'." Vance extended his match-case to Arnesson, who had been filling his pipe as he talked. "Was Pardee well acquainted with Sprigg?"

    "Oh, no. Met him here twice—that's all. Pardee knows Drukker well, though. Always asking him about potentials and scalars and vectors. Hopes to hit on something that'll revolutionize chess."

    "Was he interested in the Riemann-Christoffel tensor when you discussed it the other night?"

    "Can't say that he was. A bit out of his realm. You can't hitch the curvature of space-time to a chess-board."

    "What do you make of this formula being found on Sprigg?"

    "Don't make anything of it. If it had been in Sprigg's handwriting I'd say it dropped out of his pocket. But who'd go to the trouble of trying to type a mathematical formula?"

    "The Bishop apparently."

    Arnesson took his pipe from his mouth and grinned.

    "Bishop X. We'll have to find him. He's full of whimsies. Perverted sense of values."

    "Obviously." Vance spoke languidly. "And, by the by, I almost forgot to ask you: does the Dillard house harbor any revolvers?"

    "Oho!" Arnesson chuckled with unrestrained delight. "Sits the wind there? . . . Sorry to disappoint you. No revolvers. No sliding doors. No secret stairways. All open and above-board."

    Vance sighed theatrically.

    "Sad . . . sad! And I had such a comfortin' theory."

    Belle Dillard had come silently down the hall, and now stood in the archway. She had evidently heard Vance's question and Arnesson's answer.

    "But there are two revolvers in the house, Sigurd," she declared. "Don't you remember the old revolvers I used for target practice in the country?"

    "Thought you'd thrown 'em away long ago." Arnesson rose and drew up a chair for her. "I told you when we returned from Hopatcong that summer that only burglars and bandits are allowed to own guns in this benevolent State. . . ."

    "But I didn't believe you," the girl protested. "I never know when you're jesting and when you're serious."

    "And you kept them, Miss Dillard?" came Vance's quiet voice.

    "Why—yes." She shot an apprehensive glance at Heath. "Shouldn't I have done so?"

    "I believe it was technically illegal. However"—Vance smiled reassuringly—"I don't think the Sergeant will invoke the Sullivan law against you.—Where are they now?"

    "Down-stairs—in the archery-room. They're in one of the drawers of the tool-chest."

    Vance rose.

    "Would you be so good, Miss Dillard, as to show us where you put them? I have a gnawin' curiosity to see 'em, don't y' know."

    The girl hesitated and looked to Arnesson for guidance. When he nodded she turned without a word and led the way to the archery-room.

    "They're in that chest by the window," she said.

    Going to it she drew out a small deep drawer in one end. At the rear, beneath a mass of odds and ends, was a .38 Colt automatic.

    "Why!" she exclaimed. "There's only one here. The other is gone."

    "It was a smaller pistol, wasn't it?" asked Vance.

    "Yes. . . ."

    "A .32?"

    The girl nodded and turned bewildered eyes on Arnesson.

    "Well, it's gone, Belle," he told her, with a shrug. "Can't be helped. Probably one of your young archers took it to blow out his brains with after he'd foozled at shooting arrows up the alley."

    "Do be serious, Sigurd," she pleaded, a little frightened. "Where could it have gone?"

    "Ha! Another dark mystery," scoffed Arnesson. "Strange disappearance of a discarded .32."

    Seeing the girl's uneasiness Vance changed the subject.

    "Perhaps, Miss Dillard, you'd be good enough to take us to Mrs. Drukker. There are one or two matters we want to speak to her about; and I assume, by your presence here, that the ride in the country has been postponed."

    A shadow of distress passed over the girl's face.

    "Oh, you mustn't bother her to-day." Her tone was tragically appealing. "Lady Mae is very ill. I can't understand it—she seemed so well when I was talking with her up-stairs. But after she'd seen you and Mr. Markham she changed: she became weak and . . . oh, something terrible seemed to be preying on her mind. After I'd put her to bed she kept repeating in an awful whisper: 'Johnny Sprig, Johnny Sprig.' . . . I phoned her doctor and he came right over. He said she had to be kept very quiet. . . ."

    "It's of no importance," Vance assured her. "Of course we shall wait.—Who is her doctor, Miss Dillard?"

    "Whitney Barstead. He's attended her as long as I can remember."

    "A good man," nodded Vance. "There's no better neurologist in the country. We'll do nothing without his permission."

    Miss Dillard gave him a grateful look. Then she excused herself.

    When we were again in the drawing-room Arnesson stationed himself before the fireplace and regarded Vance satirically.

    "'Johnny Sprig, Johnny Sprig.' Ha! Lady Mae got the idea at once. She may be cracked, but certain lobes of her brain are over-active. Unaccountable piece of machinery, the human brain. Some of the greatest mental computers of Europe are morons. And I know a couple of chess masters who need nurses to dress and feed 'em."

    Vance appeared not to hear him. He had stopped by a small cabinet near the archway and was apparently absorbed in a set of jade carvings of ancient Chinese origin.

    "That elephant doesn't belong there," he remarked casually, pointing to a tiny figure in the collection. "It's a bunjinga—decadent, don't y' know. Clever, but not authentic. Probably a copy of a Manchu piece." He stifled a yawn and turned toward Markham. "I say, old man, there's nothing more we can do. Suppose we toddle. We might have a brief word with the professor before we go, though. . . . Mind waiting for us here, Mr. Arnesson?"

    Arnesson lifted his eyebrows in some surprise, but immediately crinkled his face into a disdainful smile.

    "Oh, no. Go ahead." And he began refilling his pipe.

    Professor Dillard was much annoyed at our second intrusion.

    "We've just learned," said Markham, "that you were speaking to Mrs. Drukker before breakfast this morning. . . ."

    The muscles of Professor Dillard's cheeks worked angrily.

    "Is it any concern of the District Attorney's office if I speak to a neighbor in my garden?"

    "Certainly not, sir. But I am in the midst of an investigation which seriously concerns your house, and I assumed that I had the privilege of seeking help from you."

    The old man spluttered a moment.

    "Very well," he acquiesced irritably. "I saw no one except Mrs. Drukker—if that's what you're after."

    Vance projected himself into the conversation. "That's not what we came to you for, Professor Dillard. We wanted merely to ask you if Mrs. Drukker gave you the impression this morning that she suspected what had taken place earlier in Riverside Park."

    The professor was about to make a sharp retort, but checked himself. After a moment he said simply:

    "No, she gave me no such impression."

    "Did she appear in any way uneasy or, let us say, excited?"

    "She did not!" Professor Dillard rose and faced Markham. "I understand perfectly what you are driving at and I won't have it. I've told you, Markham, that I'll take no part in spying or tale-bearing where this unhappy woman is concerned. That's all I have to say to you." He turned back to his desk. "I regret I'm very busy to-day."

    We descended to the main floor and made our adieus to Arnesson. He waved his hand to us cordially as we went out; but his smile held something of contemptuous patronage, as if he had witnessed, and was gloating over, the rebuff we had just received.

    When we were on the sidewalk Vance paused to light a fresh cigarette.

    "Now for a brief causerie with the sad and gentlemanly Mr. Pardee. I don't know what he can tell us, but I have a yearnin' to commune with him."

    Pardee, however, was not at home. His Japanese servant informed us that his master was most likely at the Manhattan Chess Club.

    "To-morrow will be time enough," Vance said to Markham, as we turned away from the house. "I'll get in touch with Doctor Barstead in the morning and try to arrange to see Mrs. Drukker. We'll include Pardee in the same pilgrimage."

    "I sure hope," grumbled Heath, "that we learn more to-morrow than we did to-day."

    "You overlook one or two consolin' windfalls, Sergeant," returned Vance. "We've found out that every one connected with the Dillard house was acquainted with Sprigg and could easily have known of his early morning walks along the Hudson. We've also learned that the professor and Mrs. Drukker were ramblin' in the garden at eight o'clock this morning. And we discovered that a .32 revolver has disappeared from the archery-room.—Not an embarrassment of riches, but something—oh, decidedly something."

    As we drove down-town Markham roused himself from gloomy abstraction, and looked apprehensively at Vance.

    "I'm almost afraid to go on with this case. It's becoming too sinister. And if the newspapers get hold of that Johnny-Sprig nursery rhyme and connect the two murders, I hate to think of the gaudy sensation that'll follow."

    "I fear you're in for it, old man," sighed Vance. "I'm not a bit psychic—never had dreams that came true, and don't know what a telepathic seizure feels like—but something tells me that the Bishop is going to acquaint the press with that bit of Mother-Goose verse. The point of his new joke is even obscurer than his Cock-Robin comedy. He'll see to it that no one misses it. Even a grim humorist who uses corpses for his cap-and-bells must have his audience. Therein lies the one weakness of his abominable crimes. It's about our only hope, Markham."

    "I'll give Quinan a ring," said Heath, "and find out if anything has been received."

    But the Sergeant was saved the trouble. The World reporter was waiting for us at the District Attorney's office, and Swacker ushered him in at once.

    "Howdy, Mr. Markham." There was a breezy impudence in Quinan's manner, but withal he showed signs of nervous excitement. "I've got something here for Sergeant Heath. They told me at Headquarters that he was handling the Sprigg case, and said he was parleying with you. So I blew over."

    He reached in his pocket and, taking out a sheet of paper, handed it to Heath. "I'm being mighty high, wide and handsome with you, Sergeant, and I expect a little inside stuff by way of reciprocity. . . . Cast your eye on that document. Just received by America's foremost family journal."

    It was a plain piece of typewriting paper, and it contained the Mother-Goose melody of Johnny Sprig, typed in élite characters with a pale-blue ribbon. In the lower right-hand corner was the signature in capitals: THE BISHOP.

    "And here's the envelope, Sergeant." Quinan again dug down into his pocket.

    The official cancellation bore the hour of 9 a.m., and, like the first note, it had been mailed in the district of Post Office Station "N."